342 XV NORMED ALGEBRAS AND SPECTRAL THEORY (b) Deduce from (a) that, if we put x' -- continuous involution on A such that (#')* ¥> (#*)' for some x e A. 12. Let E be a Hilbert space. Show that a self-adjoint compact operator U is a Hilbert- Schmidt operator if and only if, (Art) being the sequence of eigenvalues of U eacl counted according to its multiplicity, the sum]T |An|2 is finite. The sum is then equa to ||ffHI. 13. (a) The norm ||«||2 is defined as in (15.4.8) when E is a finite-dimensional Hilber space. In a Hilbert space of dimension n, give an example of an operator it such thai < 1 2 (b) Let E be a Hilbert space of infinite dimension. Give an example of a se- quence (un) of Hilbert-Schmidt operators on E such that the sequence of numbers l|w?«ir||2/WI! tends to 0 as n tends to oo. 14. Let X be a (metrizable, separable) locally compact space, and let ft be a positive measure on X. Let K(JC, y) be a function on X x X, belonging to &c$i x X, ft ® ft). For each function /e &c(X, ft), the function x\-*\ K(x,y)f(y) d^(y) is defined al- most everywhere, and its class belongs to L£(X, ft) (use the Lebesgue-Fubini theorem and the Cauchy-Schwarz inequality). If this class is denoted by U •/ show that U is a Hilbert-Schmidt operator; U is said to be associated to the kernel K. Show that II U\\2 = N2(K) and that U* is associated to the kernel (x, y)^K(y,x). If Ul9 U2 are associated to kernels KI, K2 respectively, belonging to ^c(X x X, ft (x) ft), then Ui U2 is associated to the kernel K(*^) = 15. Let E be an infinite-dimensional Hilbert space and U a Hilbert-Schmidt operator on E. If A is a regular value for £/, show that the operator (U— A • IE)"* is of the form V— A"1 * 1E> where Kis a Hilbert-Schmidt operator. Deduce that the spectrum of U in the algebra -Sf(E) is the same as in the algebra obtained by adjoining a unit element to &2(E). For every function / which is analytic in an open set containing Sp(£7), the operator f(U) (Section 15.2, Problem 11) is a Hilbert-Schmidt operator. 16. Let HI, ..., Hn be self-adjoint operators, each pair of which commute, on a separable Hilbert space E, and suppose that \\ffj\( ^ 1 and that \\HjHk\\ ^£IJ~"1 for ally, k, where e is a real number such that 0 g e < 1. Then ||#i + • • • + -ft|| ^ (1 + \/fi)/(l - e) (" Collar's lemma "). (Using the Gelfand-Neumark theorem, reduce to proving the same property for real numbers ult.,., un. For each t i> 0, let v(t) be the number of indices/ such that \uj\>t. Remark that Jo and prove that v(t) :g 1 + [2 log //log e], by observing that if \uj\e£t and \uk\ iy = 0,